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Logarithmics Differentiation Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Logarithmics Differentiation

We next show how to use the logarithm as an aid to differentiation. The key idea is that if F is a function taking positive values then we can exploit the formula

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Example 1

Calculate the derivative of the function

F ( x ) = (cos x ) (sin x ) , 0 < x < .

Solution 1

We take the natural logarithm of both sides:

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Now we calculate the derivative using the formula (*) preceding this example: The derivative of the left side of (†) is

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Using the product rule, we see that the derivative of the far right side of (†) is

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

We conclude that

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Thus

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

You Try It : Differentiate log 9 |cos x |.

You Try It : Differentiate 3 sin(3 x ) . Differentiate x sin 3 x .

Example 2

Calculate the derivative of F ( x ) = x 2 · (sin x ) · 5 x .

Solution 2

We have

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

Using formula (*), we conclude that

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

 hence

Transcendental Functions 6.4 Calculus with Logs and Exponentials to Arbitrary Bases

You Try It : Calculate ( d/dx )[(ln x ) ln x ].

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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